Question

What is the the vertex of `x = (y -3)^2 - 9`?

Answer 1

The vertex coordinates are (3, -9).

Explanation:

Let us consider that the variables were inverted on purpose. That way, y is the horizontal axis and x is the vertical one.

First of all, solve the Mathematical Identitiy:
`(y-3)^2=(y-3)*(y-3)=y^2-3y-3y+9`
Then simplify the function:
`x=y^2-3y-3y-9+9=y^2-6y`

From this point on, there are many ways to find the vertex. I prefer the one that doesn't use formulas. Every quadratic formula takes the shape of a parabola, and every parabola has a symmetry axis. That means points that have the same height have the same distance from the center. Therefore, let's calculate the roots:

`y(y-6)=0`
`y'=0`
`y''->y-6=0`
`y''=6`

Find the point that is between the roots: `(0+6)/2=3`. Therefore, `yv=3`. Now, to find the x value corresponding, just solve the function for 3:
`x(3)=(3)^2-6*(3)=9-18=-9`.

Therefore, the axis is located at (3, -9).
graph{(x-3)^2-9 [-2, 8, -10, 10]}